Wedge-shaped HfO_(2) buffer layer-induced field-free spin-orbit torque switching of HfO_(2)/Pt/Co structure

Field-free spin-orbit torque(SOT)switching of perpendicular magnetization is essential for future spintronic devices.This study demonstrates the field-free switching of perpendicular magnetization in an HfO_(2)/Pt/Co/TaO_(x) structure,which is facilitated by a wedge-shaped HfO_(2)buffer layer.The field-free switching ratio varies with HfO_(2)thickness,reaching optimal performance at 25 nm.This phenomenon is attributed to the lateral anisotropy gradient of the Co layer,which is induced by the wedge-shaped HfO_(2)buffer layer.The thickness gradient of HfO_(2)along the wedge creates a corresponding lateral anisotropy gradient in the Co layer,correlating with the switching ratio.These findings indicate that field-free SOT switching can be achieved through designing buffer layer,offering a novel approach to innovating spin-orbit device.

Bessel vortices in spin-1 Bose-Einstein condensates with Zeeman splitting and spin-orbit coupling

We investigate the ground states of spin-orbit coupled spin-1 Bose-Einstein condensates in the presence of Zeeman splitting.By introducing the generalized momentum operator,the linear version of the system is solved exactly,yielding a set of Bessel vortices.Additionally,based on linear solution and using variational approximation,the solutions for the full nonlinear system and their ground state phase diagrams are derived,including the vortex states with quantum numbers m=0,1,as well as mixed states.In this work,mixed states in spin-1 spin-orbit coupling(SOC)BEC are interpreted for the first time as weighted superpositions of three vortex states.Furthermore,the results also indicate that under strong Zeeman splitting,the system cannot form localized states.The variational solutions align well with numerical simulations,showing stable evolution and meeting the criteria for long-term observation in experiments.

Spin-orbit torque effect in silicon-based sputtered Mn_(3)Sn film

Noncollinear antiferromagnet Mn_(3)Sn has shown remarkable efficiency in charge-spin conversion,a novel magnetic spin Hall effect,and a stable topological antiferromagnetic state,which has resulted in great interest from researchers in the field of spin-orbit torque.Current research has primarily focused on the spin-orbit torque effect of epitaxially grown noncollinear antiferromagnet Mn_(3)Sn films.However,this method is not suitable for large-scale industrial preparation.In this study,amorphous Mn_(3)Sn films and Mn_(3)Sn/Py heterostructures were prepared using magnetron sputtering on silicon substrates.The spin-torque ferromagnetic resonance measurement demonstrated that only the conventional spin-orbit torque effect generated by in-plane polarized spin currents existed in the Mn_(3)Sn/Py heterostructure,with a spin-orbit torque efficiency of 0.016.Additionally,we prepared the perpendicular magnetized Mn_(3)Sn/CoTb heterostructure based on amorphous Mn_(3)Sn film,where the spin-orbit torque driven perpendicular magnetization switching was achieved with a lower critical switching current density(3.9×10^(7)A/cm^(2))compared to Ta/CoTb heterostructure.This research reveals the spin-orbit torque effect of amorphous Mn_(3)Sn films and establishes a foundation for further advancement in the practical application of Mn_(3)Sn materials in spintronic devices.

Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides

The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem.

The symplectic eigenfunction expansion theorem and its application to the plate bending equation

This paper deals with the bending problem of rectangular plates with two opposite edges simply supported. It is proved that there exists no normed symplectic orthogonal eigenfunction system for the associated infinite-dimensional Hamiltonian operator H and that the two block operators belonging to Hamiltonian operator H possess two normed symplectic orthogonal eigenfunction systems in some space. It is demonstrated by using the properties of the block operators that the above bending problem can be solved by the symplectic eigenfunction expansion theorem, thereby obtaining analytical solutions of rectangular plates with two opposite edges simply supported and the other two edges supported in any manner.

Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity

The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved. The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.

Solving ground eigenvalue and eigenfunction of spheroidal wave equation at low frequency by supersymmetric quantum mechanics method

The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.

Skeletons of 3D Surfaces Based on the Laplace-Beltrami Operator Eigenfunctions

In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.

Eigenfunction expansion method and its application to two-dimensional elasticity problems based on stress formulation

This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectie eigenfunction expansion method.

EIGENFUNCTIONS OF THE NONLINEAR ELLIPTIC EQUATION WITH CRITICAL EXPONENT IN R^2

We consider the following eigenvalue problem: [GRAPHICS] Where f(x, t) is a continuous function with critical growth. We prove the existence of nontrivial solutions.

A NEW METHOD FOR ESTABLISHING PSEUDO ORTHOGONAL PROPERTIES OF EIGENFUNCTION EXPANSION FORM IN FRACTURE MECHANICS

A new and simple method is developed to establish the pseudo orthogonal properties (POP) of the eigenfunction expansion form (EEF) of crack-tip stress complex potential functions for cracked anisotropic and piezoelectric materials, respectively. Di?erent from previous research, the complex argument separation technique is not required so that cumbersome manipulations are avoided. Moreover, it is shown, di?erent from the previous research too, that the orthogonal properties of the material characteristic matrices A and B are no longer necessary in obtaining the POP of EEF in cracked piezoelectric materials. Of the greatest signi?cance is that the method presented in this paper can be widely extended to treat many kinds of problems concerning path- independent integrals with multi-variables.

Slip flow in an annular sector duct using radial eigenfunctions

The fully developed slip flow in an annular sector duct is solved by expansions of eigenfunctions in the radial direction and boundary collocation on the straight sides. The method is efficient and accurate. The flow field for slip flow differs much from that of no-slip flow. The Poiseuille number increases with increased inner radius, opening angle, and decreases with slip.

A complete symplectic eigenfunction expansion for the elastic thin plate with simply supported edges

The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated.First,the problem for a rectangular plate with simply supported edges is solved directly.Then,the completeness of the eigenfunctions is proved,thereby demonstrating the feasibility of using separation of variables to solve the problem. Finally,the general solution is obtained by using the proved expansion theorem.

An Alternative Method of Eigenfunction Expansion Associated with Second Order Differential Equation in Infinite Domain and Its Application

In this paper a method of eigenfunction expansion associated with 2nd order differential equation is developed by using the concept of theory of distribution. An application of the method to the infinite long antenna is described in detail.

Ground eigenvalue and eigenfunction of a spin-weighted spheroidal wave equation in low frequencies

Spin-weighted spheroidal wave functions play an important role in the study of the linear stability of rotating Kerr black holes and are studied by the perturbation method in supersymmetric quantum mechanics. Their analytic ground eigenvalues and eigenfunctions are obtained by means of a series in low frequency. The ground eigenvalue and eigenfunction for small complex frequencies are numerically determined.

Eigenfunction expansion method of upper triangular operator matrixand application to two-dimensional elasticity problems based onstress formulation

This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.

Asymptotic Approximation of the Eigenvalues and the Eigenfunctions for the Orr-Sommerfeld Equation on Infinite Intervals

Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, Wentzel-Kramers-Brillouin (WKB) methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms of Green’s functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which the eigenvalues and the eigenfunctions can be approximated. In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions. Our procedure works regardless of the values of the Reynolds number.

Characterization of Periodic Eigenfunctions of the Fourier Transform Operator

We generalize this result to p1,p2-periodic eigenfunctions of F?on R2 and to p1,p2,p3-periodic eigenfunctions of F?on R3.

Vectorial spin-orbital Hall effect of light upon tight focusing and its experimental observation in azopolymer films

Hall effect of light is a result of symmetry breaking in spin and/or orbital angular momentum(OAM)possessing optical system and is caused by e.g.refractive index gradient/interface between media or focusing of a spatially asymmetrical beam,similar to the electric field breaking the symmetry in spin Hall effect for electrons.The angular momentum(AM)conservation law in the ensuing asymmetric system dictates redistribution of spin and orbital angular momentum,and is manifested in spin-orbit,orbit-orbit,and orbit-spin conversions and reorganization,i.e.spin-orbit and orbit-orbit interaction.This AM restructuring in turn requires shifts of the barycenter of the electric field of light.In the present study we show,both analytically and by numerical simulation,how different electric field components are displaced upon tight focusing of an asymmetric light beam having OAM and spin.The relation between field components shifts and the AM components shifts/redistribution is presented too.Moreover,we experimentally demonstrate,for the first time,to the best of our knowledge,the spin-orbit Hall effect of light upon tight focusing in free space.This is achieved using azopolymers as a media detecting longitudinal or z component of the electrical field of light.These findings elucidate the Hall effect of light and may broaden the spectrum of its applications.